Optimization Methods and Software
Vol. 00, No. 00, January 2008, 1–14
RESEARCH ARTICLE
Extreme Diffusion Values for non-Gaussian Diffusions
Deren Hana , Liqun Qib∗ and Ed X. Wu
c
a
Institute of Mathematics, School of Mathematics and Computer Science, Nanjing
Normal University, Nanjing, Jiangsu, China; b Department of Applied Mathematics, The
Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong; c Department of
Electrical and Electronic Engineering, The University of Hong Kong, Hong Kong.
(27 September 2007; revised 6 July 2008; in final form 7 July 2008 )
A new Magnetic Resonance Imaging (MRI) model, called Diffusion Kurtosis Imaging (DKI),
was recently proposed, to characterize the non-Gaussian diffusion behavior in tissues. DKI
involves a fourth order three dimensional tensor and a second order three dimensional tensor.
Similar to those in the Diffusion Tensor Imaging (DTI) model, the extreme diffusion values
and extreme directions associated to this tensor pair play important roles in DKI. In this
paper, we study the properties of the extreme values and directions associated to such tensor
pairs. We also present a numerical method and its preliminary computational results.
Keywords: Diffusion Kurtosis Tensors, Extreme Diffusion Values, Extreme Diffusion
Directions, Anisotropy.
1.
Introduction
Magnetic resonance imaging in tissues has been used to infer anatomical structure
and to aid in the diagnosis of many pathologies [11, 13]. Nowadays, the most
successful and popular magnetic resonance (MR) technique is the diffusion tensor
imaging, which uses a second order tensor D to quantify a diffusion anisotropy [3, 8].
When the diffusion process is Gaussian, the MR signal attenuates exponentially as
a function of b-value, i.e.,
ln(S(b)) = ln(S(0)) − bDapp ,
(1)
where
2
Dapp = Dx =
3
X
i,j=1
∗ Corresponding
author. Email: [email protected].
ISSN: 1055-6788 print/ISSN 1029-4937 online
c 2008 Taylor & Francis
°
DOI: 10.1080/1055678xxxxxxxxxxxxx
http://www.informaworld.com
Dij xi xj ,
(2)
2
Deren Han, Liqun Qi and Ed X. Wu
is the apparent diffusion coefficient (ADC) along
the gradient direction x =
P
(x1 , x2 , x3 ) with components xi , i = 1, 2, 3 and 3i=1 x2i = 1,
µ
2
b = (γδg)
δ
∆−
3
¶
and g is the gradient strength, γ is the proton gyromagnetic ratio, δ is a pulse duration, ∆ is a time interval between the centers of the diffusion sensitizing gradient
pulse, and D is a symmetric second order tensor with elements Dij , i, j = 1, 2, 3.
The success of DTI is based on the assumption that water molecules obey Gaussian diffusion in biological tissues. In reality, we often meet diffusions that are
non-Gaussian in the confining environment of biological tissues, causing that the
DTI model breaks down [1, 3]. For example, when DTI is used in regions where
the fibers cross or merge, difficulty is often encountered since with current MR
resolution, voxel averaging of different fiber tracts is frequent and unavoidable.
To overcome this problem, new MR imaging models [2, 9, 14] have been proposed,
which use higher order tensors, rather than just a second order tensor used in DTI,
to characterize the process of diffusion. One of such new MR imaging models is
diffusion kurtosis imaging [6, 10]. In that model, a fourth order three dimensional
fully symmetric tensor, called the diffusion kurtosis (DK) tensor, is proposed to
describe the non-Gaussian behavior of water molecules in tissues. That is, it is
assumed that the MR signal attenuates as a function of b-value in the following
way,
1
2
ln(S(b)) = ln(S(0)) − bDapp + b2 Dapp
Kapp ,
6
(3)
where Kapp is the apparent kurtosis coefficient (AKC) along x,
Kapp =
2
MD
W x4 ,
2
Dapp
3
X
W x4 ≡
(4)
Wijkl xi xj xk xl ,
i,j,k,l=1
and
MD =
D11 + D22 + D33
3
is the mean diffusivity.
For the DTI model, Pierpaoli and Basser [15] pointed out “The most intuitive and
simplest rotationally invariant indices are ratios of the principal diffusivities, such
as the dimensionless anisotropy ratio λ1 /λ3 that measures the relative magnitudes
of the diffusivities along the fiber-tract direction and one transverse direction.” In
DKI, the D-eigenvalues of W and the D-eigenvector associated with these eigenvalues also play important roles. They describe the extreme AKC values and the
extreme deviations of the diffusion from Gaussian diffusion, and are invariant under
rotations of the co-ordinate systems [19, 21]. However, some important properties
in the DKI model need to be studied further. For example, which direction is the
fastest/slowest diffusion direction in the DKI model? How can we measure the
Extreme diffusion values for non-Gaussian diffusions
3
anisotropy of the tissue? To answer these questions, we have to find the extreme
points associated to the diffusion tensor D and the diffusion kurtosis tensor W
together. In this paper, we study these problems and propose a numerical method
to find such extreme points. We also present some numerical examples to illustrate
the method.
2.
Notation and Preliminary Results
We use the notation in [4, 12, 16–19] for the tensors and vectors. We use x =
(x1 , x2 , x3 )T to denote the direction vector, which is denoted as n = (n1 , n2 , n3 )T
in [6, 10]. According the result of [6], the ADC and AKC for a single direction
should satisfy the relationship (3), i.e.,
1
2
ln[S(b)] = ln[S(0)] − bDapp + b2 Dapp
Kapp .
6
(5)
D is a second order tensor and W is a fourth order tensor, whose elements are
obtained by filling experimental data into equation (5) and solving the resulting
system of linear equations by singular value decomposition or least squares methods. Let the eigenvalues of D be α1 ≥ α2 ≥ α3 . Then the mean diffusivity [3] can
be calculated by
MD =
α1 + α2 + α3
.
3
In the DTI model, one assumes that the diffusion obeys a Gaussian distribution
and there is no quadratic term in (5), i.e., the ADC for a single direction should
satisfy the relationship (1)
ln[S(b)] = ln[S(0)] − bDapp .
(6)
In this case, the directions of the fastest and the slowest diffusion are eigenvectors
associated to the largest and the smallest eigenvalues of the second order tensor
D, which can be obtained via solving the optimization problems
max Dx2
s.t. xT x = 1,
(7)
min Dx2
s.t. xT x = 1,
(8)
and
respectively. For the DKI model, the study in [19] was focused on the properties
of W which can be used to measure the deviation of the diffusion from a Gaussian
one. For example, the AKC value is used to measure the average deviation; the
largest and smallest D-eigenvalues of the fourth order tensor W , defined as
max W x4
s.t. Dx2 = 1,
(9)
4
Deren Han, Liqun Qi and Ed X. Wu
and
min W x4
s.t. Dx2 = 1,
(10)
can be used to measure the largest and the smallest deviation from Gaussian diffusion and the associated eigenvectors are the fastest and the slowest deviation
directions.
In a similar way as in the DTI model, we are now going to find the fastest and the
slowest diffusion values and the associated diffusion directions of water molecules
in the tissue, under a non-Gaussian diffusion that has relationship (5). That is, we
need to solve the following optimization problems
2 W x4
max Dx2 − 16 bMD
T
s.t.
x x = 1,
(11)
2 W x4
min Dx2 − 16 bMD
T
s.t.
x x = 1.
(12)
and
The solutions of (11) and (12) depend on the second order tensor D and the fourth
order tensor W . Thus, our tasks are to find some useful properties of solutions
of (11) and (12), the extreme values and the associated extreme directions of a
tensor pair (D, W ), and to design numerical methods for finding such values and
directions.
It is known that Dx is a vector in <3 with its ith component as
(Dx)i =
3
X
Dij xj ,
j=1
for i = 1, 2, 3. As in [16–19], we denote W x3 as a vector in <3 with its ith component
as
3
(W x )i =
3
X
Wijkl xj xk xl ,
j,k,l=1
for i = 1, 2, 3. Without loss of generality, we assume that D is positive definite.
Then α1 ≥ α2 ≥ α3 > 0. In practice, this assumption is natural, as the ADC value
should be positive in general.
3.
Properties of the Extreme Values
The critical points of problems (11) and (12) satisfy the following equation for
some λ ∈ <:
½
2 W x3 = λx,
Dx − 13 bMD
T
x x = 1.
(13)
Extreme diffusion values for non-Gaussian diffusions
5
2 W . Then (13) can be rewritten as
Let W̄ = 13 bMD
½
Dx − W̄ x3 = λx,
xT x = 1.
(14)
A real number λ satisfying (13) with a real vector x is called an extreme diffusion
value of the non-Gaussian diffusion, and the real vector x associated to λ is called
an extreme diffusion direction.
The following theorem shows the existence of the extreme diffusion values.
Theorem 3.1 The extreme diffusion values always exist. If x is a solution of (14)
associated with an extreme diffusion value λ, then
λ = Dx2 − W̄ x4 .
(15)
The largest diffusion value is equal to λmax , and the smallest diffusion value is
equal to λmin .
Proof. The feasible regions of (11) and (12) are compact and their objective
functions are continuous. Hence, each of these two optimization problems has at
least one solution, which must satisfy (14) with corresponding Lagrangian multipliers. Hence, the largest diffusion value and the smallest diffusion value always
exist and (15) follows from the fact (14) directly. This completes the proof.
¤
The following theorem shows an important property of the extreme diffusion
values.
Theorem 3.2 The extreme diffusion values of a non-Gaussian diffusion are invariant under rotations of coordinate systems.
Proof. With a rotation, x, D and W̄ are converted to y = P x, D̂ = DP 2 , and
Ŵ = W̄ P 4 , respectively. Here, P = (pij ) is the rotation matrix and the elements
of D̂ and Ŵ are defined by
3
X
D̂ij =
0
Di0 j 0 pi0 i pj 0 j ,
0
i ,j =1
and
3
X
Ŵijkl =
0
0
0
W̄i0 j 0 k0 l0 pi0 i pj 0 j pk0 k pl0 l ,
0
i ,j ,k ,l =1
see [16] for the definition of orthogonal similarity. If λ is an extreme diffusion value
with an extreme direction x, then we have
½
D̂y − Ŵ y 3 = λy,
y T y = 1,
indicating that λ is still an extreme diffusion value in the new coordinate system. Thus, extreme diffusion values of non-Gaussian diffusion are invariant under
rotations of coordinate systems.
¤
6
4.
Deren Han, Liqun Qi and Ed X. Wu
A Method for Finding the Extreme Points
To find the extreme diffusion values in DKI, we need to solve optimization problems
(11) and (12), which are optimization problems with polynomial objective functions
and constraints. The first-order optimal conditions for (11) and (12) are system of
polynomial equations (13). For solving this system of polynomial equations, we
can use Groebner bases and resultants in elimination theory, see [5, 20]. However,
using such methods directly to (13) may be time consuming. Moreover, the final
one variable equation derived from (13) may have higher degree, which makes it
sensitive to the coefficients.
In the following, we propose a direct method to solve (13), which fully uses the
structure of the problem. The first step is to eliminate λ from the system and then
uses the last equation to eliminate x3 from the system. Finally, it solves a system
of polynomial equations with two variables, adopting the method of resultants.
Note that Theorem 3.2 indicates that we may rotate the co-ordinate system
such that the three orthogonal eigenvectors of D are used as the co-ordinate base
vectors. In that co-ordinate system, the representative matrix of D is a diagonal
matrix. Therefore, we may assume that


α1 0 0
D =  0 α2 0  ,
0 0 α3
which implies that Ŵ = W̄ . Consequently, (14) can be written as
Ŵ1111 x31 + 3Ŵ1112 x21 x2 + 3Ŵ1113 x21 x3 + 3Ŵ1122 x1 x22 + 6Ŵ1123 x1 x2 x3
+3Ŵ1133 x1 x23 + Ŵ1222 x32 + 3Ŵ1223 x22 x3 + 3Ŵ1233 x2 x23 + Ŵ1333 x33 = (α1 − λ)x1 ,
Ŵ2111 x31 + 3Ŵ1122 x21 x2 + 3Ŵ1123 x21 x3 + 3Ŵ1222 x1 x22 + 6Ŵ1223 x1 x2 x3
+3Ŵ1233 x1 x23 + Ŵ2222 x32 + 3Ŵ2223 x22 x3 + 3Ŵ2233 x2 x23 + Ŵ2333 x33 = (α2 − λ)x2 ,
Ŵ1113 x31 + 3Ŵ1123 x21 x2 + 3Ŵ1133 x21 x3 + 3Ŵ1223 x1 x22 + 6Ŵ1233 x1 x2 x3
+3Ŵ1333 x1 x23 + Ŵ2223 x32 + 3Ŵ2233 x22 x3 + 3Ŵ2333 x2 x23 + Ŵ3333 x33 = (α3 − λ)x3 ,
x21 + x22 + x23 = 1.
(16)
Note that the coefficients in the above equations come from the fact that the
tensor Ŵ is symmetric, i.e., its entries Ŵijkl are invariant under any permutation
of their indices i, j, k and l.
To find the extreme diffusion values and the associated extreme diffusion directions, we have to solve the above system of polynomial equations on x1 , x2 , x3 and
λ. For this system of equations, we have the following result.
Theorem 4.1 We have the following results on the extreme diffusion values and
their associated extreme diffusion directions.
(a). If Ŵ1112 = Ŵ1113 = 0, then λ = α1 − Ŵ1111 is an extreme diffusion values of the
non-Gaussian diffusion with the extreme diffusion direction x = (1, 0, 0)> .
(b). For any real roots t of the following equations:

 Ŵ1112 t4 − (Ŵ1111 − 3Ŵ1122 − α1 + α2 )t3 − 3(Ŵ1112 − Ŵ1222 )t2
−(3Ŵ1122 − Ŵ2222 − α1 + α2 )t − Ŵ1222 = 0,

Ŵ1113 t3 + 3Ŵ1123 t2 + 3Ŵ1223 t + Ŵ2223 = 0,
(17)
Extreme diffusion values for non-Gaussian diffusions
7
λ = Dx2 − Ŵ x4
is an extreme diffusion values with the corresponding extreme diffusion direction
x = ±√
1
(t, 1, 0)> .
1 + t2
(18)
1
(u, v, 1)>
+ v2 + 1
(19)
(c). λ = Dx2 − Ŵ x4 and
x = ±√
u2
constitute an extreme diffusion values and extreme diffusion direction pair, where
u and v are real solutions of the following system of polynomial equations

Ŵ1113 u4 + 3Ŵ1123 u3 v − (Ŵ1111 − 3Ŵ1133 − α1 + α3 )u3 + 3Ŵ1223 u2 v 2




−(3Ŵ1112 − 6Ŵ1233 )u2 v − 3(Ŵ1113 − Ŵ1333 )u2





+(3Ŵ2233 − 3Ŵ1122 − α3 + α1 )uv 2



3

 +Ŵ2223 uv − (6Ŵ1123 − 3Ŵ2333 )uv − (3Ŵ1133 − Ŵ3333 − α1 + α3 )u


−Ŵ1222 v 3 − 3Ŵ1223 v 2 − 3Ŵ1233 v − Ŵ1333 = 0,
(20)

Ŵ1113 u3 v − Ŵ1112 u3 + 3Ŵ1123 u2 v 2 − (3Ŵ1122 − 3Ŵ1133 − α2 + α3 )u2 v




−3Ŵ1123 u2 + 3Ŵ1123 uv 3 − (3Ŵ1222 − 6Ŵ1233 )uv 2




−(6Ŵ1223 − 3Ŵ1333 )uv − 3Ŵ1233 u − 3(Ŵ2223 − Ŵ2333 )v 2





+Ŵ2223 v 4 − (Ŵ2222 − 3Ŵ2233 − α2 + α3 )v 3


−(3W2233 − α2 − Ŵ3333 + α3 )v − Ŵ2333 = 0.
All the extreme diffusion values and the associated directions are given by (a),
(b) and (c) if Ŵ1112 = Ŵ1113 = 0, and by (b) and (c) otherwise.
Proof. It is direct to check that (a) holds.
Setting x3 = 0, x2 6= 0 and using the third equation in (16), we have

(Ŵ1111 + α1 )x31 + 3Ŵ1112 x21 x2 + (3W̄ 1122 + α1 )x1 x22 + Ŵ1222 x32 = λx1 ,



Ŵ2111 x31 + (3Ŵ1122 + α2 )x21 x2 + 3W̄1222 x1 x22 + (Ŵ2222 + α2 )x32 = λx2 ,
3
2
2
3


 Ŵ21113 x21 + 3Ŵ1123 x1 x2 + 3Ŵ 1223x1 x2 + Ŵ2223 x2 = 0,
x1 + x2 = 1.
Let t = x1 /x2 . Then from the first three equations, we have (17) and from the last
equation we have (18). This proves (b).
8
Deren Han, Liqun Qi and Ed X. Wu
If x3 6= 0, then from the fourth equation in (16), we have

3
2
2
2

 (Ŵ1111 + α1 )x1 + 3Ŵ1112 x1 x2 + 3Ŵ1113 x1 x3 + (3Ŵ 1122 + α1 )x1 x2


+6Ŵ1123 x1 x2 x3 + (3Ŵ1133 + α1 )x1 x23 + Ŵ1222 x32





+3Ŵ1223 x22 x3 + 3Ŵ1233 x2 x23 + Ŵ1333 x33 = λx1 ,




Ŵ2111 x31 + (3Ŵ1122 + α2 )x21 x2 + 3W̄1123 x21 x3 + 3Ŵ1222 x1 x22 + 6Ŵ1223 x1 x2 x3



+3Ŵ1233 x1 x23 + (Ŵ2222 + α2 )x32 + 3Ŵ2223 x22 x3

+(3Ŵ2233 + α2 )x2 x23 + Ŵ2333 x33 = λx2 ,




Ŵ1113 x31 + 3Ŵ1123 x21 x2 + (3W̄1133 + α3 )x21 x3 + 3Ŵ 1223x1 x22 + 6Ŵ1233 x1 x2 x3





+3Ŵ1333 x1 x23 + Ŵ2223 x32 + (3Ŵ2233 + α3 )x22 x3




Ŵ2333 x2 x23 + (Ŵ3333 + α3 )x33 = λx3 ,

 +3
2
x1 + x22 + x23 = 1.
(21)
Let u = x1 /x3 and v = x2 /x3 . Then (c) follows immediately from the above system
of equations.
¤
To find all the extreme diffusion values and the corresponded diffusion directions
for non-Gaussian diffusion, from Theorem 4.1, we need to solve systems of equations
(17) and (20). (17) is a system of polynomial equations of one variable t, which
can be solved efficiently. (20) is a system of polynomial equations of two variables
u and v. For solving such equations, we first regard it as a system of polynomial
equations of variable u and rewrite it as
½
γ0 u4 + γ1 u3 + γ2 u2 + γ3 u + γ4 = 0,
τ0 u3 + τ1 u2 + τ2 u + τ3 = 0,
where γ0 , · · · , γ4 , τ0 , · · · , τ3 are polynomials of v, which can be calculated by (20).
The above system of polynomial equations in u possesses solutions if and only if
its resultant vanishes [5]. The resultant of this system of polynomial equations is
the determinant of the following 7 × 7 matrix

γ0
0

0

V := 
 τ0
0

0
0
γ1
γ0
0
τ1
τ0
0
0
γ2
γ1
γ0
τ2
τ1
τ0
0
γ3
γ2
γ1
τ3
τ2
τ1
τ0
γ4
γ3
γ2
0
τ3
τ2
τ1
0
γ4
γ3
0
0
τ3
τ2

0
0

γ4 

0
,
0

0
τ3
which is a polynomial equation in variable v. After finding all real roots of this
polynomial, we can substitute them to (20) to find all the real solutions of u. Correspondingly, all the extreme diffusion values and the associated diffusion directions
can be found.
5.
Algorithm Description
We now give our algorithm for solving (13).
Algorithm. A Direct Algorithm for (13)
Extreme diffusion values for non-Gaussian diffusions
9
Input: The second-order diffusion tensor D, the fourth-order kurtosis
tensor W and the b value.
Output: The extreme diffusion values and the associated diffusion
directions.
S1. Find the decomposition of D = P ΛP T , where Λ is a diagonal matrix whose
diagonal elements are eigenvalues of D, α1 ≥ α2 ≥ α3 > 0, and P is a orthogonal
matrix whose columns are eigenvectors of D.
2 W, where
S2. Let W̄ = 13 bMD
MD =
α1 + α2 + α3
3
and Ŵ = W̄ P 4 , i.e.,
3
X
Ŵijkl =
0
0
0
W̄i0 j 0 k0 l0 pi0 i pj 0 j pk0 k pl0 l .
0
i ,j ,k ,l =1
S3. Let
g(v) := det V,
where V is the 7 × 7 matrix defined by (4) and find the zeros of g(v).
S4. For every real zeros vi found at the above step, substitute it into (17) to find the
solution uj .
S5. From each pair of vi and uj found at the above two steps, form the extreme
diffusion directions and the extreme diffusion values.
¤
6.
Numerical Examples
In this section, we report some computational results on the extreme diffusion
values and the associated diffusion directions of a second order and a fourth order
tensor pair that derived from data of MRI experiments on rat spinal cord specimen
fixed in formalin. The MRI experiments were conducted on a 7 Tesla MRI scanner
at Laboratory of Biomedical Imaging and Signal Processing at The University of
Hong Kong.
In MRI experiments, the AKC and ADC values for a given gradient x ∈ R3 can
be determined by acquiring data at three or more b values [6] including b=0. In
our experiments, we take six b values 0, 800, 1600, 2400, 3200 and 4000, in unit of
s/mm2 . In each example, we take 30 gradient directions and get the corresponding
AKC and ADC values as the averages of the 9 pixels. From these ADC and AKC
values, we obtain the elements of the diffusion tensor D and the diffusion kurtosis
tensor W by using the least squares method, discussed in [6] and [10].
Example 1. Our first example is taken from the white matter. The diffusion tensor
D is


0.1755 0.0035 0.0132
D =  0.0035 0.1390 0.0017  ∗ 10−3
0.0132 0.0017 0.4006
10
Deren Han, Liqun Qi and Ed X. Wu
in unit of mm2 /s. The eigen-decomposition of the diffusion tensor D is D̂ =
DP 2 , where D̂ is a diagonal matrix whose diagonal elements are (α1 , α2 , α3 ) =
(0.4013, 0.1751, 0.1387) ∗ 10−3 and


0.0584 0.9939 0.0938
P =  0.0073 0.0935 −0.9956  .
0.9983 −0.0589 0.0018
The fifteen independent elements of the diffusion kurtosis tensor W are W1111 =
0.4982, W2222 = 0, W3333 = 2.6311, W1112 = −0.0582, W1113 = −1.1719, W1222 =
0.4880, W2223 = −0.6162, W1333 = 0.7639, W2333 = 0.7631, W1122 = 0.2236,
W1133 = 0.4597, W2233 = 0.1519, W1123 = −0.0171, W1223 = 0.1852 and W1233 =
−0.4087, respectively. It is easy to find that
µ
2
MD
=
D11 + D22 + D33
3
¶2
= 5.6813 × 10−8 .
To find the largest and the smallest diffusion values, we need first obtain the largest
and the smallest D-eigenvalues. For given b value, we can use the method proposed
in Section 4 to compute all the extreme diffusion values and the associated diffusion
directions. Table 1 lists the results for b = 2400 (s/mm2 ).
Table 1. Extreme diffusion values and directions of (D, W ).
1
2
3
4
5
6
7
8
9
10
11
x1
-0.5400
0.1487
-0.2051
-0.6674
0.7043
-0.9957
0.0319
0.0466
-0.5908
0.6435
-0.5459
x2
-0.8413
-0.9656
0.9251
0.6483
-0.5416
-0.0675
0.5815
-0.4651
-0.3367
0.2619
0.1195
x3
0.0258
0.2135
0.3195
0.3665
0.4589
0.0632
0.8129
0.8840
0.7332
0.7192
0.8293
λ × 103
0.2483
0.1499
0.1438
0.2212
0.2795
0.3288
0.1531
0.1278
0.2206
0.2170
0.2183
From the above table we can see that the largest and the smallest diffusion values
for this example are 0.3288 × 10−3 and 0.1278 × 10−3 (mm2 /ms), attained at
(−0.9957, −0.0675, 0.0632)>
and (0.0466, −0.4651, 0.8840)> ,
respectively.
To show the dependence of the extreme diffusion values on the b values, we plot
the largest and the smallest diffusion values as functions of the b values. Figure 1
shows the result, where the y-axis is scaled to 103 .
To give some insight to the difference between the DTI and DKI, we also plot
the largest diffusion values in these two models, as functions of b values, and the
result is Figure 2.
Figure 2 clearly shows that when b is too small, the linear model (6) can model
the diffusion behavior quite well; while as b value becomes larger, the difference
between the two models (5) and (6) is more obvious.
Extreme diffusion values for non-Gaussian diffusions
11
5
Extrme Diffusion Values ln(s(b)/s(0))
Largest
4
Smallest
3
2
1
0
1000
2000
3000
4000
2
b value (s/mm )
Figure 1. Largest and Smallest Diffusion Values as functions of b values.
2
DKI
DKI Vs DTI
DTI
1
0
0
1000
2000
3000
4000
2
b value (s/mm )
2 K
Figure 2. Largest bDapp VS bDapp − 16 b2 Dapp
app as functions of b.
Example 2. Our second example is taken from the gray matter. The diffusion
tensor D is


1.2455 −0.0169 −0.0012
D =  −0.0169 1.6921 0.0077  ∗ 10−3
−0.0012 0.0077 1.1937
12
Deren Han, Liqun Qi and Ed X. Wu
in unit of mm2 /s. The eigen-decomposition of the diffusion tensor D is D̂ =
DP 2 , where D̂ is a diagonal matrix whose diagonal elements are (α1 , α2 , α3 ) =
(1.6928, 1.2448, 1.1936) ∗ 10−3 and


0.0379 −0.9991 0.0174
P =  −0.9992 −0.0381 −0.0148  .
−0.0154 0.0168 0.9997
The fifteen independent elements of the diffusion kurtosis tensor W are W1111 =
0.1171 × 10−5 , W2222 = 0.2665 × 10−5 , W3333 = 0.1425 × 10−5 , W1112 = −0.0009 ×
10−5 , W1113 = 0.0031 × 10−5 , W1222 = 0.0026 × 10−5 , W2223 = 0.0046 × 10−5 ,
W1333 = 0.0044 × 10−5 , W2333 = −0.0008 × 10−5 , W1122 = 0.0456 × 10−5 , W1133 =
0.0348 × 10−5 , W2233 = 0.0681 × 10−5 , W1123 = 0.0016 × 10−5 , W1223 = −0.0015 ×
10−5 and W1233 = 0.0013 × 10−5 , respectively. We can find that
µ
2
MD
=
D11 + D22 + D33
3
¶2
= 1.8964 × 10−6 .
For b = 2400 (s/mm2 ), we can use the method proposed in Section 4 to compute
all the extreme diffusion values and the associated diffusion directions. Table 2 lists
the results.
Table 2. Extreme diffusion values and directions of (D, W ).
1
2
3
4
5
6
7
8
x1
-0.4485
0.6697
0.6697
-0.0000
-1.0000
0.7070
-0.7070
0.0000
x2
0.8938
-0.7426
0.7426
1.0000
-0.0000
0.0002
-0.0001
-0.0001
x3
0
0.0001
0.0001
0.0001
0.0000
0.7072
0.7072
1.0000
λ × 103
1.3349
1.4457
1.4457
1.2448
1.6926
1.4430
1.4430
1.1934
From the above table we can see that the largest and the smallest diffusion values
for this example is 1.6926 × 10−3 and 1.1934 × 10−3 (mm2 /ms), attained at
(−1.0000, −0.0000, 0.0000)>
and (0.0000, −0.0001, 1.0000)> ,
respectively.
To show the dependence of the extreme diffusion values on the b values, we list
the largest and the smallest diffusion values for different b values in the following
table (scaled to 103 ).
Table 3. Extreme diffusion values of (D, W ).
b
Largest
Smallest
800
1.4425
1.1932
1600
1.4419
1.1928
2400
1.4412
1.1925
3200
1.4406
1.1921
4000
1.4399
1.1917
Table 3 shows that both the largest and the smallest diffusion values are decreasing function of the b value; however, the speed to decrease is not as clear as the first
example. The reason is that in the second example, the elements of the diffusion
Extreme diffusion values for non-Gaussian diffusions
13
kurtosis tensor W is too small, comparing to those of the diffusion tensor D. In
other words, the diffusion in the second example is more likely to be a Gaussian
diffusion.
To give some insight to the difference between the DTI and DKI, we also plot
the largest diffusion values in these two models, as functions of b values, and the
result is Figure 3.
7000
DKI
DTI
6000
5000
4000
3000
2000
1000
0
0
1000
2000
3000
4000
2
b Value (s/mm )
2 K
Figure 3. Largest bDapp VS bDapp − 16 b2 Dapp
app as functions of b.
7.
Final Remarks
In this paper, we proposed the extreme diffusion values and the associated diffusion
directions, which are the extreme values and the extreme points associated to the
diffusion tensor and the diffusion kurtosis tensor. We analyzed some properties of
the extreme diffusion values and proposed a numerical method for finding such
values and the associated directions. These values and directions are potentially
useful for understanding tissue microstructure.
It is believed that noise will be of greater effects on the solution because higher
diffusion gradients are used in DKI and the least squares method is used for estimating the fourth order tensor, W . The effects of Rician noise will be likely similar
to those in the case of diffusion tensor imaging, as studied in [7]. A study on such
a noise effect will be a future further work.
Acknowledgments. The authors wish to thank Professor Oleg Burdakov for
his encouragement, Professor Regina Burachik and two referees for their comments
and help. This work was supported by the Natural Science Foundation of China
(Grant No.10501024), the Natural Science Foundation of Jiangsu province (Grant
No. BK2006214), the Hong Kong Research Grant Council and a Chair Professor
Fund of The Hong Kong Polytechnic University.
14
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